Here’s a puzzle based on a passing remark in Andrew Hodges’ biography of Alan Turing.
Hodges says that in 1927, Alan Turing, then 15 years old, discovered the Taylor series for arctangent “starting from the trigonometrical formula for tan(x/2).”
Deriving the series could be a homework exercise in a calculus class, but Turing discovered the series without using calculus.
Hodge doesn’t give any further information about what Turing’s derivation. What might it have been? How could you derive the series expansion for arctangent from a trig identity?